Existence and uniqueness of minimal blow-up solutions to an inhomogeneous mass critical NLS

We consider the 2-dimensional focusing mass critical NLS with an inhomogeneous nonlinearity: i ₜu+ u+k (x) |u|^2u=0. From a standard argument, there exists a threshold Mₖ>0 such that H¹ solutions with \|u\|₋ℂMₖ. In this paper, we consider the dynamics at threshold \|u₀\|₋ℂ=Mₖ and give a necessary and sufficient condition on k to ensure the existence of critical mass finite time blow-up elements. Moreover, we give a complete classification in the energy class of the minimal finite time blow-up elements at a nondegenerate point, hence extending the pioneering work by Merle who treated the pseudoconformal invariant case k 1.